I've uninstalled Ubuntu One from my system (Ubuntu 12.04) but I still get the Ubuntu One right click menu item (share, get link, etc) and the folders I used to sync still show Ubuntu One's icons (synced, syncing, etc) These issues happening in Nautilus.

If you have not logged out and back in, the plug-in code is still loaded and running in Nautilus. Any application which uses plug-ins will have to be restarted after those plug-ins have been removed, before they will disappear from the interface. For Nautilus, the easiest way to do this is to simply log out, and then back in after a few seconds.

Nope, restarted the machine and the integration is still there.
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GabrielAug 22 '12 at 17:07

Then you have not actually uninstalled Ubuntu One yet.
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dobeyAug 23 '12 at 14:18

yes, I had uninstalled Ubuntu One at the time (I did it through the Software Center). Restarting the system did nothing. @green7's answer was what made the Nautilus integration go away.
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GabrielAug 23 '12 at 19:16

You probably only removed the control panel, and not all of Ubuntu One, which is several applications and libraries that integrate with the system. Restarting the system didn't work, because you only removed one piece of the Ubuntu One suite; the control panel.
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dobeyAug 24 '12 at 1:31

As I said, I removed the Ubuntu One entry in the Software Center, which should have been enough since it's what any regular user would do. It wasn't. To remove it completely I had to apply green7's answer which shouldn't have to be necessary. It should've gone away after uninstalling, it didn't. I shouldn't have had the need to use the terminal but I did.
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GabrielAug 24 '12 at 10:20